On the spectral asymptotics and summability by the Abel method of series in a system of root vector functions of nonsmooth elliptic differential operators that are far from selfadjoint. (Q1432281)

The author considers the differential operator \(A_0 \equiv \sum_{| \alpha| \leq 2m}a_{\alpha} D^{\alpha}\) acting in a Sobolev space of vector valued functions on an open domain \(\Omega\) of \(\mathbb R^n\) with the cone property. The coefficients \(a_{\alpha}\) are assumed to be continuous and with certain summability properties. The author assumes conditions on the location of the eigenvalues of a closed extension \(A\) of \(A_{0}\) which prevent \(A\) from being self-adjoint, and proves an asymptotic formula for the distribution of the eigenvalues of \(A\) by exploiting an approximation argument and techniques that the author had already developed for operators with smooth coefficients. Then the author considers a differential operator of order \(2m\) generated by a quadratic form with essentially bounded coefficients and proves an estimate for the resolvent and an asymptotic formula for the distribution of the eigenvalues, which refines previous work of the author and of S. A. Iskhakov, and obtains a statement of summability of a Fourier expansion associated to \(A\) with the Abel method.